Nonlocal Chemical Bond

Going to the non-local level of the softness kernel one relevant choice should be shaped as (Garza Robles, 1993 Putz, 2008b, d) 

In these conditions, the chemical action equation for non-local or delocalized bonding description is raised as  

Worth remarking that the explicit bonding displacement Ar modulates the chemical action amplitude as Ar vanishes as the previous local case of bonding is recovered, see the previous case. Nevertheless, both local and non-local instants of chemical action produce the present chemical bonding picture in a complete non-orbital way. More details and discussions are in next addressed. 

Since the non-local bonding equation contains the local case it may be rearranged to the convenient form recalling a sort of adapted Heisenberg relation for chemical bonding the present discussion follows (Putz, 2009a,l)  

Finally, aiming to find the explicit localized-delocalized bonding density one has to integrate the adapted Heisenberg equation further rewritten as (Putz, 2009a,f) 

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Putz, M. V. (2010b). Beyond quantum nonlocality chemical bonding field. Int J. Environ. Sci. 1 25-31. 

Physical hardness can be defined to be proportional, and sometimes equal, to the chemical hardness (Parr and Yang, 1989). The relationship between the two types of hardness depends on the type of chemical bonding. For simple metals, where the bonding is nonlocal, the bulk modulus is proportional to the chemical hardness density. The same is true for non-local ionic bonding. However, for covalent crystals, where the bonding is local, the bulk moduli may be less appropriate measures of stability than the octahedral shear moduli. In this case, it is also found that the indentation hardness—and therefore the Mohs scratch hardness—are monotonic functions of the chemical hardness density. 

We note that in the course of dealing with the fifth issue, the introduction of nonlocal chemical responses, we have successfully addressed the third issue, the definition of isoelectronic reactivity indices for both localized and extended systems. We note also that recognition of the need for a nonlocal description of chemical responses came early. Huckel s bond-bond polarizability [49] is in fact a simplified version of P(r, r ) [3]. 

Let us apply the idea of the Cahn-Hilliard approach to a diblock copolymer, where (pA and 4>b are now the reduced local densities of monomers A and B which are chemically bonded in the diblock-copolymer hnear chain molecule. As before, we shall assume that 4>a) = a—4>b i 4>) = 0) as the order parameter. It has been shown [33]-[35] that the long-range interaction of monomers in a copolymer chain can be described by an additional nonlocal term in the Ginzburg-Landau free energy functional ... 

The functional transformation of electronic density to energy is not known, but good approximations can be obtained using the generalized gradient approximation (GGA) functionals [82-84]. The DFT/GGA methods are quite successful in treating chemical bonds, since they account for nonlocal electron correlation. 

Calculated DFT properties listed in Table 1 were obtained from the fit of the ground-state potential energy curves to 12 points calculated around the energy minimum [32]. Dissociation energy has been corrected for basis set superposition error by a standard counterpoise technique. The local approximation to the exchange and correlation gives the best fit to bond distances, theoretical values differ by no more than 0.03 A (4%) from the experimental ones (see Table 1). Vibrational frequencies are also predicted to lie within 1 % off the experiment. One should remember, however, that other advanced quantum chemical methods give equally satisfactory results for these, basicaly one-electron quantities and that inclusion of nonlocal effects does not improve the DFT predictions. The dipole moment, fi, is much more sensitive... 

He emphasized that quantum chemistry was built upon many chemical concepts, often inherited from classical chemical theories. K-electrons, L-electrons, valence electrons, 7c- or (7-electrons, localized or delocalized electrons, bonds, and so forth, were certainly incompatible with the quantum mechanical ideas of indistinguishability and nonlocality but had proved to be very useful tools for chemists and had been appropriated by quantum chemists (Daudel 1952). Hence, he argued, they should be given alternative formulations compatible with the new physics. Such was the case of the concept of "loge" introduced by Daudel and his students to explain quantum-... 

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